Question

Weary of the low turnout in student elections, a college administration decides to choose an SRS of three students to form an advisory board that represents student opinion. Suppose that 48% of all students oppose the use of student fees to fund student interest groups, and that the opinions of the three students on the board are independent. Then the probability is 0.48 that each opposes the funding of interest groups.Call the three students A, B, and C. Give the probability of each possible outcomes (AcBcCc are the events that the students support the student fee usage).Pr(ABC) = _____Pr(ABCc) = _____Pr(ABcC) = _____Pr(AcBC) = _____Pr(ABcCc) = _____Pr(AcBCc) = _____Pr(AcBcC) = _____Pr(AcBcCc) = _____

Answer 1

`P(ABC) = 0.110592`

`P(ABC^c) = 0.119808`

`P(AB^cC) = 0.119808`

`P(A^cBC) = 0.119808`

`P(AB^cC^c) = 0.129792`

`P(A^cBC^c) = 0.129792`

`P(A^cB^cC) = 0.129792`

`P(A^cB^cC^c) = 0.140608`

Explanation:

Given

`P(A) = P(B) = P(C) = 48\%`

Convert the probability to decimal

`P(A) = P(B) = P(C) = 0.48`

Solving (a): P(ABC)

This is calculated as:

`P(ABC) = P(A) * P(B) * P(C)`

This gives:

`P(ABC) = 0.48*0.48*0.48`

`P(ABC) = 0.110592`

Solving (b):
`P(ABC^c)`

This is calculated as:

`P(ABC^c) = P(A) * P(B) * P(C^c)`

In probability:

`P(C^c) = 1 - P(C)`

So, we have:

`P(ABC^c) = P(A) * P(B) * (1 - P(C))`

`P(ABC^c) = 0.48 * 0.48 * (1 - 0.48)`

`P(ABC^c) = 0.48 * 0.48 * 0.52`

`P(ABC^c) = 0.119808`

Solving (c):
`P(AB^cC)`

This is calculated as:

`P(AB^cC) = P(A) * P(B^c) * P(C)`

`P(AB^cC) = P(A) * [1 - P(B)] * P(C)`

`P(AB^cC) = 0.48 * (1 - 0.48)* 0.48`

`P(AB^cC) = 0.48 * 0.52* 0.48`

`P(AB^cC) = 0.119808`

Solving (d):
`P(A^cBC)`

This is calculated as:

`P(A^cBC) = P(A^c) * P(B) * P(C)`

`P(A^cBC) = [1-P(A)] *P(B) * P(C)`

`P(A^cBC) = (1 - 0.48)* 0.48 * 0.48`

`P(A^cBC) = 0.52* 0.48 * 0.48`

`P(A^cBC) = 0.119808`

Solving (e):
`P(AB^cC^c)`

This is calculated as:

`P(AB^cC^c) = P(A) * P(B^c) * P(C^c)`

`P(AB^cC^c) = P(A) * [1-P(B)] * [1-P(C)]`

`P(AB^cC^c) = 0.48 * [1-0.48] * [1-0.48]`

`P(AB^cC^c) = 0.48 * 0.52*0.52`

`P(AB^cC^c) = 0.129792`

Solving (f):
`P(A^cBC^c)`

This is calculated as:

`P(A^cBC^c) = P(A^c) * P(B) * P(C^c)`

`P(A^cBC^c) = [1-P(A)] * P(B) * [1-P(C)]`

`P(A^cBC^c) = [1-0.48] * 0.48 * [1-0.48]`

`P(A^cBC^c) = 0.52 * 0.48 * 0.52`

`P(A^cBC^c) = 0.129792`

Solving (g):
`P(A^cB^cC)`

This is calculated as:

`P(A^cB^cC) = P(A^c) * P(B^c) * P(C)`

`P(A^cB^cC) = [1-P(A)] * [1-P(B)] * P(C)`

`P(A^cB^cC) = [1-0.48] * [1-0.48] * 0.48`

`P(A^cB^cC) = 0.52 * 0.52 * 0.48`

`P(A^cB^cC) = 0.129792`

Solving (h):
`P(A^cB^cC^c)`

This is calculated as:

`P(A^cB^cC^c) = P(A^c) * P(B^c) * P(C^c)`

`P(A^cB^cC^c) = [1-P(A)] * [1-P(B)] * [1-P(C)]`

`P(A^cB^cC^c) = [1-0.48] * [1-0.48] * [1-0.48]`

`P(A^cB^cC^c) = 0.52*0.52*0.52`

`P(A^cB^cC^c) = 0.140608`